Why the CAPM Cross-Section Adds N GMM Moment Conditions

# How come the cross-sectional CAPM equation produces $N$ moment conditions (not $1$)?


# How come the cross-sectional CAPM equation produces $N$ moment conditions (not $1$)?












Reading Cochrane "Asset Pricing" (2005) section 12.2 (p. 241), I got lost in the derivation of the GMM estimator for the single-factor model. Equation $(12.23)$ says the moments are $$ g_T(b) = \begin{bmatrix} E(R^e_t-a-\beta f_t) \\ E[(R^e_t-a-\beta f_t)f_t] \\ E(R^e-\beta \lambda) \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \tag{12.23} $$ where $R^e_t=(R^e_{1,t},\dots,R^e_{N,t})'$ is a vector of individual assets' excess returns, $\beta=(\beta_1,\dots,\beta_N)'$ is a vector of betas, $f_t$ is factor's excess return and $\lambda=E(f)$ is the expected value of the factor's excess return. The first two rows correspond to time series regressions for $N$ assets (one regression per asset), so there are actually $2N$ conditions. If I understand correctly, the third row corresponds to a cross-sectional regression of time-averaged returns. It is a single equation (scalar dependent variable) $$ E_T(R^{ei})=\beta_i' \lambda+\alpha_i, \quad i=1,2,\dots,N. \tag{12.10} $$ ($(12.10)$ is specified for potentially many factors, but $(12.23)$ considers the simple case of a single factor, so vector $\beta'$ turns into scalar $\beta$, and the same holds for $\lambda$.)

That should add one moment condition. However, the textbook says it adds $N$ moment conditions. How does that happen?

A related question is "GMM estimation of the CAPM: why not include sample mean of the market excess return as a moment?".

## Answer by Richard Hardy (score 1)

https://quant.stackexchange.com/a/74540

This is simple. While $(12.10)$ is indeed a single equation, the third row of $(12.23)$ combines $N$ copies of $(12.10)$, one of each asset. Thus, $R^e$ and $\beta$ are both vectors of $N$ elements, and the result is $N$ conditions.